Einstein Relatively Easy

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As we will see later, the Bianchi Identity equation will be of fundamental importance to find the Einstein equation.

Also the complete, unalterated form of the Riemann curvature tensor doesn't appear in the Einstein field equations. Instead, it is contracted to give two other important measures of the curvature known as the Ricci tensor and the Ricci scalar.

In this article, our aim is to define these three important Rieman tensor derivatives.

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If you like this content, you can help maintaining this website with a small tip on my tipeee page  

 

As we will see later, the Bianchi Identity equation will be of fundamental importance to find the Einstein equation.

Also the complete, unalterated form of the Riemann curvature tensor doesn't appear in the Einstein field equations. Instead, it is contracted to give two other important measures of the curvature known as the Ricci tensor and the Ricci scalar.

In this article, our aim is to define these three important Rieman tensor derivatives.

The Bianchi Identity

The Bianchi Identity

 First let us try to demonstrate the Bianchi Identity

From our previous article The Riemann curvature tensor part III: Symmetries and independant components, we know that at the origin of a Local Inertial Frame (LIF), we have:

We also know  from Introduction to Covariant Differentiation that at the origin of Local Inertial Frame, the Christoffel symbols do all vanish, and then the covariant derivative becomes the ordinary derivative:

Therefore, we get, at the origin of a LIF:

By cyclically permuting the index of the derivative with the last two indices of the tensor, we get:

By adding up these the three equations and using the commutativity of partial derivatives, we see that the terms cancel in pairs, so we get what we want

The Ricci tensor and Ricci scalar

The Ricci tensor and Ricci scalar

First, we can contract the first and third indices of the Riemann curvature tensor to get the Ricci tensor

Using both the Riemann tensor and metric symmetries, we show easily that the Ricci tensor itself is symmetric 

so the Ricci tensor is symmetric.

We can contract the Ricci tensor in turn to get the curvature scalar or Ricci scalar R

 

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"The essence of my theory is precisely that no independent properties are attributed to space on its own. It can be put jokingly this way. If I allow all things to vanish from the world, then following Newton, the Galilean inertial space remains; following my interpretation, however, nothing remains.."
Letter from A.Einstein to Karl Schwarzschild - Berlin, 9 January 1916

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice."
Einstein to Max Born, letter 52, 4th december 1926

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